In math literature on functional analysis I found various short-hand notations for specific types of convergence, e.g. a single right arrow for pointwise convergence
$$f_n(x) \underset{n \to \infty}{\to} f(x)$$
as opposed to paired arrows for uniform convergence
$$f_n(x) \underset{n \to \infty}{⇉} f(x)$$
Are these notations commonly known and accepted? Are there any standard notations to abbreviate clumsy constructs such as
$$ \forall \varepsilon>0 \exists N \forall x \forall n>N: |f_n(x)-f(x)|<\varepsilon $$